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In this paper, using improvement-valued maps, we define two types of E-Benson proper efficient elements for subsets within a linear space under a variable ordering map C. Consequently, we delve into studying two types of E-Benson proper efficient solutions of vector optimization problems under variable ordering structures. We establish relationships among different types of E-Benson proper efficient elements. Furthermore, we demonstrate that the two types of E-Benson proper efficiency, in relation to the ordering map C, not only unify and extend certain notions of (weakly) nondominated elements but also extend some well-known notions of Benson proper efficiency under fixed ordering structures. Lastly, under suitable assumptions, we establish linear scalarization theorems for E-Benson proper efficient solutions of vector optimization problems under variable ordering structures. Several examples are also provided to illustrate the derived results. [ABSTRACT FROM AUTHOR]

This document is an editorial from the Journal of Nonlinear & Variational Analysis, dedicated to commemorating the life and achievements of Prof. Dr. Habil. Alfred Göpfert. It provides a brief biography of Prof. Göpfert, highlighting his contributions to mathematics, academia, and research. The editorial also mentions his impact on students and colleagues, his prolific publications, and his expertise in convex analysis and optimization theory. The special issue includes twelve articles that cover various topics in mathematics and optimization, showcasing the legacy and inspiration of Prof. Göpfert. [Extracted from the article]

In this paper, we state that generalized convexity, invex sets, and the Fr'echet differentiability assumption associated with curvilinear integral type functionals represent some necessary and sufficient mathematical tools for establishing various connections between the solutions of some robust (weak) vector commanded variational inequalities and (weak, proper) robust efficient solutions of the corresponding multi-objective variational control problem. In addition, the physical motivation of the problem under investigation is formulated in the illustrative application given at the end of this paper. [ABSTRACT FROM AUTHOR]

Outer approximation (OA) for solving convex mixed-integer nonlinear programming (MINLP) problems is heavily dependent on the convexity of functions and a natural issue is to relax the convexity assumption. This paper is devoted to OA for dealing with a pseudo-convex MINLP problem. By solving a sequence of nonlinear subproblems, we use Lagrange multiplier rules via Clarke subdifferentials of subproblems to introduce a parameter and then equivalently reformulate such MINLP as the mixed-integer linear program (MILP) master problem. Then, an OA algorithm is constructed to find the optimal solution to the MNILP by solving a sequence of MILP relaxations. The OA algorithm is proved to terminate after a finite number of steps. Numerical examples are illustrated to test the constructed OA algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we study the Cauchy problem for a semilinear fractional order differential inclusion with a nonconvex-valued almost lower semicontinuous nonlinearity and a linear closed operator generating a C0-semigroup in a separable Banach space. By using the theory of measure of noncompactness and condensing operators, we study topological properties of the solution set of this problem. We prove that the solution set of the Cauchy problem possesses the classical Kneser connectedness property. [ABSTRACT FROM AUTHOR]

To deal with nondifferentiable interval-valued functions (IVFs) (not necessarily convex), we present the notion of Fréchet subdifferentiability or gH-Fréchet subdifferentiability. We explore its relationship with gH-differentiability and develop various calculus results for gH-Fréchet subgradients of extended IVFs. By using the proposed notion of subdifferentiability, we derive new necessary optimality conditions for unconstrained interval optimization problems (IOPs) with nondifferentiable IVFs. We also provide a necessary condition for unconstrained weak sharp minima of an extended IVF in terms of the proposed notion of subdifferentiability. Examples are pesented to support the main results. [ABSTRACT FROM AUTHOR]

In this paper, a forward-backward splitting algorithm with two inertial parameters (one nonnegative and the other non-positive) extrapolation step is proposed for finding a zero point of the sum of maximal monotone and co-coercive operators in real Hilbert spaces. One of the interesting features of our proposed algorithm is that no online rule on the inertial parameters with the iterates is needed. The weak convergence result of the proposed algorithm is established under some standard assumptions. Numerical results arising from LASSO problems in compressed sensing, image processing, and SCAD penalty problems are provided to illustrate the behavior of our proposed algorithm. [ABSTRACT FROM AUTHOR]

An introduction is presented in which the editor discusses articles in the issue on topics including image restoration methods, optimization algorithms, and the integration of traditional and deep learning approaches.

The class of quasimonotone variational inequalities is more general and applicable than the class of pseudomonotone and monotone variational inequalities. However, few results can be found in the literature on quasimonotone variational inequalities and currently results are mostly on weak convergent methods in the framework of Hilbert spaces. In this paper, we study the class of non-Lipschitz quasimonotone variational inequalities and the class of non-Lipschitz variational inequalities without monotonicity in the framework of Banach spaces. We propose a new inertial Tseng’s extragradient method and obtain some strong convergence results for the proposed algorithm under some mild conditions on the control parameters. While the cost operator is non-Lipschitz, our proposed method does not require any linesearch procedure but employs a more efficient and simple self-adaptive step sizes with known parameters. Finally, we present several numerical experiments to demonstrate the implementability of our proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonmonotone gradient method for smooth constrained multiobjective optimization problems. Under mild assumptions, we demonstrate the Pareto stationarity of the accumulation point of the sequence generated by this method, while the convergence of the full sequence to a weak Pareto optimal solution of the problem is proven when the function is convex. Further, by imposing some assumptions on the gradients of the objective functions and the search directions, the linear convergence of the function value sequence to the optimal value is provided. The initial point in the convergence results established here can be any one in the constraint set. [ABSTRACT FROM AUTHOR]

This document is an editorial from the Journal of Nonlinear & Variational Analysis, focusing on a special issue dedicated to mathematical optimization and its applications. The editorial highlights the importance of optimization methods in various fields such as finance, engineering, and statistics, particularly in relation to machine learning. The special issue covers a range of topics, including SAR imaging, quasi-convex feasibility problems, split variational inequality problems, level-set subdifferential error bounds, sparse linear regression, non-convex minimization problems, common fixed point problems, and robust variational inequalities. The guest editors express their gratitude to the authors and reviewers who contributed to the special issue. [Extracted from the article]

Let n, k ∈ N, T > 0, and F : [0,T]×(Rn)k → 2Rn be a lower semicontinuos and bounded multifunction with nonempty closed values. We prove that there exists a bounded and upper semicontinuous multifunction G : R × (Rn)k → 2Rn with nonempty compact convex values such that every generalized solution u : [0,T] → Rn of the differential inclusion u(k) ∈ G(t,u,u',...,u(k−1)) is a generalized solution to the differential inclusion u(k) ∈ F(t,u,u',...,u(k−1)). As an application, we prove an existence and qualitative result for the generalized solutions of the Cauchy problem associated to the inclusion u(k) ∈ F(t,u,u',...,u(k−1)). In particular, we prove that if F is lower semicontinuous and bounded with nonempty closed values, then the solution multifunction admits an upper semicontinuous multivalued selection with nonempty compact connected values. Finally, by applying the latter result, we prove an analogous existence and qualitative result for the generalized solutions of the Cauchy problem associated to the differential equation g(u(k)) = f(t,u,u',...,u(k−1)), where f is continuous. We only assume that g is continuous and locally nonconstant. [ABSTRACT FROM AUTHOR]

In this paper, we aim to investigate a class of new order-theoretic properties for the resolvent operators, which are called order-preservation properties. As applications, we use these orderpreservation properties and several order-theoretic fixed point theorems to prove the existence of extremal solutions for several kinds of variational inequalities arising in mechanics and economics. In contrast to many previous studies, the approaches used in this paper are mainly based on the partial order structure of the underlying spaces, and the obtained results weaken the continuity and monotonicity of the associated mappings. [ABSTRACT FROM AUTHOR]

In classical analysis, Lebesgue first proved that R has the Lebesgue property (i.e., each Riemann integrable function from [a,b] into R is continuous almost everywhere). Though the Lebesgue property may be breakdown in many infinite dimensional spaces including Banach or quasi Banach spaces, to determine spaces with this property is still an interesting issue. This paper is devoted to the study of metrizable vector spaces with the Lebesgue property. As the main results in the paper, we prove that l¹(ܓ) (ܓ uncountable) has the Lebesgue property and Rὠ, the countable infinite product of R with itself equipped with the product topology, is a metrizable vector space with the Lebesgue property. In particular, lp, (1 < p<+∞), as a subspaces of Rɷ, is proved to have the Lebesgue property although they are Banach spaces with no such property. [ABSTRACT FROM AUTHOR]

This paper introduces several new accelerated subgradient extragradient methods with inertial effects for approximating a solution of a pseudomonotone equilibrium problem and a fixed point problem involving a quasi-nonexpansive mapping or a demicontractive mapping in real Hilbert spaces. The proposed algorithms use an adaptive non-monotonic step size criterion that does not include any Armijo line search process. Strong convergence theorems of the suggested iterative algorithms are established without the prior knowledge of the Lipschitz constants of the bifunction. Moreover, R-linear convergence is guaranteed under the assumption that the bifunction satisfies strong pseudomonotonicity. Applications to variational inequality problems are also considered. Finally, some numerical examples and applications, which demonstrate the advantages and efficiency of the proposed algorithms, are given. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonconvex constrained variational-hemivariational inequality problem for a star-shaped set. The existence of solutions is shown by an approach based on the equilibrium problem theory, and a penalization method. An application to a nonconvex constrained semipermeability model for the stationary heat conduction problem is provided. Our results are new and improve considerably recent results in literature. [ABSTRACT FROM AUTHOR]

This document is an editorial from the Journal of Applied & Numerical Optimization, announcing a special issue dedicated to Professor Yair Censor on the occasion of his 80th birthday. Professor Censor is a Professor Emeritus of Mathematics at the University of Haifa and has made significant contributions to Applied Analysis and Computational Optimization. The special issue includes contributions from researchers worldwide and covers various topics related to optimization theory, convex analysis, and numerical analysis. The editorial expresses gratitude to the authors and referees involved in the special issue and hopes that readers will enjoy it. [Extracted from the article]